Travelling from Europe
If you fly directly from Europe to Brasil you will arrive in Rio de Janeiro or São Paulo, the largest cities and hubs in Brasil. Connecting flights will take you to Manaus.
Another option is to fly via Miami in the United States which has a direct connection with Manaus.
Travelling from North America
You can fly directly to Manaus from Miami with American Airlines. Another option is to fly via Panama city. Flying via Rio de Janeiro or Sao Paulo aren't really good options, because these cities are pretty far south east of Manaus.
Travelling from other parts of Brasil
All major cities from Brasil offer direct flights to Manaus. Be aware that it is quite common that you first make one or more stopovers before arriving in Manaus. So do not exit your plane too early! The airlines that offer the majority of flights to Manaus are GOL, TAM and Azul.
Travelling by boat
If you want to be a bit more adventurous an option is also to go by boat to Manaus from Belem. It will take 5 days, but it can be an adventure in it's own right. Don't expect a lot of comfort though, unless you manage to board on one of the cruiseships.